Were-worms were creatures of an unknown kind, possibly mythical and presumably related to Dragons, that were said to dwell in the Last Desert.

Tolkien only ever mentions Were-worms once, in the quote given above, so we know almost nothing about them. We cannot even be certain that they actually existed — the Hobbits had a rich folklore full of fantastic creatures, and Were-worms quite possibly fall into that category.

Etymology

The element "were-" is a Germanic term that refers to humans. It suggests a shapeshifting creature like the werewolf of modern-day folktales. In Middle-earth, "Werewolves", such as Draugluin, did not shapechange and the name perhaps refers to the size and intelligence.

Thus Were-worms, if they did exist, could be a human-like type of Dragon, though this must remain in the realm of speculation.